# Teddy Rocks Maths Showcase 2022: Group 1 [A-B]

The first group of essays from the 2022 Teddy Rocks Maths Competition come from entrants with surnames beginning with the letters A-B. The showcase will take place throughout May and June with the winners being announced at the end.

The competition was organised with St Edmund Hall at the University of Oxford and offers a cash prize plus publication on the university website. It will be running again in early 2023 so be sure to follow Tom (InstagramTwitterFacebookYouTube) to make sure you don’t miss the announcement!

All essays can be read in full (as submitted) by clicking on the title below. If you enjoy any of them please let the author know by leaving a comment – enjoy!

Oscar introduces us to neural networks by answering the questions we’re all pondering these days, eg. how do Teslas drive themselves?

Vanessa shows us the many faces of trigonometry, including many applications you may never have realised even existed.

Zach provides a brief history of mathematics from the ancient civilisations to three titans of modern-day mathematics.

Zoe gives us a glimpse of the mathematics behind harmonious sounds and helps us to delve into the practicalities of strings and frequencies.

Majid shows us some of their favourite problems as evidence for why maths will never be obsolete, despite the rise in artificial intelligence.

Ali explores the world of topology and how it can be applied to the colour scheme in a Madeon music video.

Charlie strives to reignite our passion and imagination for mathematical proofs.

Segun explains the infamous RSA encryption algorithm and why public key cryptography is so powerful.

Gavin helps us to explore hyperbolic geometry starting with initial explorations from Sacherri.

Denim explains one of the biggest unsolved problems currently facing mathematicians and why solving it could revolutionise our lives.

1. Davide says:

I really liked “The Hyperbolic Universe” essay. I am not entirely sure about some concepts written at the beginning of the “A Hungarian Breakthrough” chapter (maybe I didn’t understand it completly).

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• Gavin Jared Bala says:

Well, it’s basically describing a geometry where the hypothesis of the obtuse angle holds, which happens to be what you get when you work on a sphere and interpret “line” as meaning the shortest path between two points that stays on the sphere – which is a great circle arc.

The triangle we’re considering has one such arc going on the equator from 0 degrees East to 90 degrees East; the next one going up the 90 degrees East meridian to the North Pole; and the last one going down the prime meridian back to where we started. The angles at the vertices are all right angles, so the angles of this triangle add up to 270 degrees. That’s more than the 180 degrees the angles of a plane triangle add up to.

On the sphere, all triangles have an angle sum of more than 180 degrees. As they get bigger, the excess is more – that is Girard’s theorem. So all quadrilaterals have to have an angle sum of more than 360 degrees, because you can break them into two triangles. In particular, a Saccheri quadrilateral has to have obtuse angles at the top, as otherwise its angle sum couldn’t exceed 360 degrees like it should.

This is interesting because it’s a familiar landscape in which the parallel postulate fails. True, there was a price: Euclid also has as another postulate that straight lines can be extended indefinitely, and we lost that here (as on the sphere they’re great circles, which curve back on themselves); but everything else works. So the fact that we can produce something like that for the obtuse angle makes the acute-angle hypothesis seem more plausible. Maybe, one might start thinking, it’s not something to be refuted. Maybe there is some surface that actualises the acute angle. In fact Lambert did suspect so.

As a matter of fact you can’t actually draw such a surface as nicely as you can with a sphere: you can embed portions of a hyperbolic plane into 3-dimensional Euclidean space, but not the whole thing (there’s got to be an edge somewhere). Such surfaces are called pseudospheres. But it helps germinate the thought that maybe the parallel postulate is really not a necessity, and that the case of the acute angle, though teeming with consequences defying common sense, might have some sort of reality as an “alternate universe” anyway. And as I wrote at the end, for all we know it might describe our own universe on the largest of scales after all (though we don’t know that of course).

Hopefully that helped!

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